Proving the bilinearity of a certain form

Question

I have a question regarding a paper:

let $φ, ψ: ℝ^3 \to ℝ$ two linear maps with $φ(v)= 〈a,v$〉 and $ψ(v)=〈b,v$〉with $a,b ∈ ℝ^3$. We define the map with $ℝ^3 \times ℝ^3 \to ℝ$ by

$$s(v,w):= φ(v) * ψ(w).$$

I need to show that s is a bilinear form and determine the transformation matrix of s regarding the canonical basis of $ℝ^3$.

I have no clue how to do this, so help would be much appreciated

Answer 1

For your second question, write your bilinear form like this

$$\underbrace{\begin{pmatrix}v_x & v_y&v_z\end{pmatrix}\begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix}}_{\langle v,a\rangle}\underbrace{\begin{pmatrix}b_x& b_y&b_z\end{pmatrix}\begin{pmatrix}v_x\\v_y\\v_z\end{pmatrix}}_{\langle b,v\rangle}=$$

$$\underbrace{\begin{pmatrix}v_x & v_y&v_z\end{pmatrix}}_{V^T}\underbrace{\begin{pmatrix}a_x b_x&a_xb_y&a_xb_z\\a_yb_x&a_yb_y&a_yb_z\\a_zb_x&a_zb_y&a_zb_z\end{pmatrix}}_M\underbrace{\begin{pmatrix}v_x\\v_y\\v_z\end{pmatrix}}_V$$

and $M$ is the transformation matrix. It is a rank one matrix because all columns of $M$ are proportional to its first column.